52.

Construct a rectangle having one of its sides 1ğ′′, and its diagonals 2" long. Make a similar rectangle having its shorter sides " long.

(S)

53. Draw two lines AB, AC, making an angle of 25° at A. Describe a circle of q′′ radius touching AB and having its centre on AC. From A draw a second tangent to the circle, marking clearly the point of contact.

(S) 54. The foci of an ellipse are 21′′ apart and its major axis is 33′′ long. Describe half the curve. (S) Construct an irregular pentagon, ABCDE, from the following

55.

data:

Sides-AB=2", BC=", AE=".

Angles-ABC and BAE each=135°. BCD and AED each=95°. Reduce the figure to a triangle of equal area having AB (produced if necessary) for base and D for vertex.

(S)

56. Draw the curve every point of which is at equal distances from a line PQ and a point F. The curve is a parabola.

(S)

57. Copy the pattern given. [180]

(A)

58. The diagram represents an incomplete scale of feet. Complete the scale so that distances of 2' may be measured by it up to 50'. The scale must be properly finished and figured. [181]

(S)

59. Make an enlarged copy of the given diagram, using the figured dimensions. [182] (S)

60. Draw two parallel lines" apart, and cut at an angle of 75° by two other parallel lines also " apart. Describe all the circles of " radius which touch any two of these lines and cut none.

N.B.-The lines are supposed to be produced indefinitely.

(S)

61. Draw two parallel lines 14′′ apart and describe a circle cutting these lines in chords respectively 1′′ and 1ğ" long.

(S)

62. From a point O equal lines OA, OB (23′′ long) are drawn including an angle of 50°. Describe the circle to which these lines are tangent at the points A, B. On OA produced determine a point such that tangents drawn to the same circle shall include an angle of 70°. (S)

63. Construct a square equal to the difference of the areas of two equilateral triangles whose sides are 3′′ and 13′′ respectively.

(S)

64. Inscribe a pentagon in a circle of 15" radius. Draw the diagonals, and the circle circumscribing the pentagon formed by their

intersections.

(S) 65. Draw a circle of 1" radius; and (a) draw a triangle circumscribing the circle, its angles being, respectively, 75°, 70°, and 35°; (6) join the points where these lines are tangent to the circle, and write down the values of the angles of the triangle thus formed. (S) 66. Draw a quadrilateral figure ABCD, with the following dimensions:

AB=2", BC=1'5′′, AD=1'5".

The diagonal BD=2′′, the diagonal AC=2'5".

Find the length of the side of a square equal in area to the quadrilateral. (S) 67. Draw a circle of 1" radius, tangent to the line AB at Q; and draw another circle touching the first circle, and tangent to AB at P. (S) 68. Construct a scale of 3.5′′ to 30', to shew feet, and correctly figured. Draw to the scale a triangle, its sides, respectively, 20', 15', and 12' long.

(S)

69. Construct a triangle ABC having AB-2", AC=175", angle CAB=40°; and draw a square equal in area to the triangle ABC.

(S) 70. A plan drawn to a scale the R.F. of which is is to be copied to a scale half as large again as the original. Construct a scale of feet and inches for the new plan (shewing 4 feet) and write above it its R.F.

(A)

71. On a French map the distance between two points known to be exactly one kilometre (32809 feet) apart measures ths of an inch, construct a comparative scale of miles and furlongs for the map and give its R.F.

(A)

72. Upon a straight line 2 inches long, construct an isosceles triangle having a vertical angle of 35°.

(A)

73. Draw a straight line, and taking a point 1 inches distant from it as centre, describe a circle of 1 inch radius. Draw a straight line touching this circle, and making an angle of 40° with the straight line.

(A)

74. Draw an equilateral triangle, of 21⁄2 inches side, and figure its points a15, 40, C35. Draw the scale of the plane containing these points, and write down its inclination to the horizontal. Unit or inch. (A)

75. A cube of 1 inch side rests with one edge in the horizontal plane, and a face containing that edge inclined at 32° to the horizontal. Draw the plan of the cube, and an elevation on a vertical plane, whose horizontal trace makes an angle of 15° with the given edge. (A)

76. The minor axis of an ellipse is 24" long, and the distance between the foci is 2". Find the major axis and draw the curve.

77. Copy the pattern. [183]

(S)

(A)

78. Draw the figure shewn, making the sides of the square

21" long. [184]

(S)

79. Construct a diagonal scale of yards, feet, and inches, R.F. 10. Shew by two small dots on the scale a distance of 8 yards 2 feet 7 inches.

(A)

80. The distance between two places on a Russian map scales 3 versts. The actual distance on the map is 2'85 inches. Construct a scale of English miles and furlongs for the map. I verst=

=3,500 feet.

(A)

81. Draw three equal circles of 1-inch radius, each touching the other two externally, and circumscribe them by another circle. (A)

82. Take any two points C and P, 2 inches apart; with centre C, radius 1 inches, describe a circle, and from P draw a straight line PQR, cutting the circle in Q and R, so that the length of the chord QR may be 1 inches. (A)

83. Inscribe a square and an equilateral triangle in a circle of 1*73 inches radius, and find by construction and write down the ratio that the area of the square inscribed in the circle bears to the area of the square described on one of the sides of the equilateral triangle.

(A)

84. On a line 1.75 inches long describe by means of your protractor, or in any way you please, a regular heptagon, and construct an equilateral triangle equal in area to 3rd of the heptagon.

(A)

85. Copy diagram [185] using dimensions shewn.

(S)

86. Copy the diagram according to the given dimensions. clearly how the centre of the small circle is determined. [186]

Shew

(S)

87. Draw the plan of a right pyramid 23′′ high, base an equilateral triangle of 2" side. The pyramid stands on its base, and the upper part is cut off by a horizontal plane 1" above the base. Indicate the section clearly by light shading.

(S)

88. Draw a triangle ABC with the following dimensions: AB=4", BC=23", AC=2". Inscribe in the triangle a parallelogram with four equal sides, one side lying on AC, and adjoining sides inclined to AC at 45°.

(S)

89. Given two lengths, AB=2′25′′, CD=2'70". Find a line whose length x is such that AB2= (x - CD) xx. Write down the value of x.

(S)

90. Draw a right-angled triangle with hypotenuse 2′′ and one side 1". Prove by construction that the square on the hypotenuse is equal to the sum of the squares on the sides.

(S)